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Bounds of Error in Higher Order Lagrange Interpolating Polynomials

Recall from the Higher Order Lagrange Interpolating Polynomials page that if $(x_0, y_0)$, $(x_1, y_1)$, …, $(x_n, y_n)$ are $n + 1$ points where $x_0$, $x_1$, …, $x_n$ are distinct numbers, then the $n^{\mathrm{th}}$ order Lagrange interpolating polynomial is given by the following formula:

The functions $L_0$, $L_1$, …, $L_n$ are given by the formula $L_k(x) = \frac{(x - x_0)(x - x_1)…(x - x_{k-1})(x - x_{k+1})…(x - x_n)}{(x_k - x_0)(x_k - x_1)…(x_k - x_{k-1})(x_k - x_{k+1})…(x_k - x_n)}$ for each $k = 0, 1, …, n$. Of course, if the $n + 1$ points above lie on the graph of $y = f(x)$ for some function $f$, then $P_n(x)$ approximates $f(x)$. Of course, $P_n(x)$ does not equal $f(x)$ and so a remainder/error term exists. The following theorem will tell us what that remainder is.